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A T-count Zero Protocol for Heralded W-state Preparation via Clifford-only Gates

Amrita Mitra

Published Jul 22, 2026• 0 citations
Publisher

Abstract

Abstract The W state is a key resource for quantum communication and distributed quantum computing. Its entanglement survives the loss of any single qubit. Preparing the W state exactly is nevertheless hard in fault-tolerant architectures. The amplitude 1/√3 lies outside the ring Z[1/√2, i]. So no Clifford+T unitary can prepare the three-qubit W state exactly, and deterministic circuits must approximate the required rotation at a T-count that grows as log(1/ϵ). We present a heralded protocol that prepares the three-qubit W state exactly using Clifford gates and a single non-Clifford operation. A shallow Clifford circuit with two CNOT gates first prepares an equal superposition of |001⟩, |010⟩, |100⟩ and |111⟩. A logical-AND measurement on one ancilla then removes the |111⟩ component without disturbing the remaining coherent superposition. The protocol succeeds with probability 3/4, a failed attempt leaves the known product state |111⟩, and each attempt consumes four T gates or one |CCZ⟩ magic state. This gives an expected cost of about 5.33 T gates independent of target precision. We prove that some non-Clifford step is necessary, and by exhaustively enumerating all 315,057,600 six-qubit stabilizer tensors we show that no post-selected Clifford circuit with three or fewer T gates outputs the W state. Every heralded Clifford+T preparation of the W state therefore uses at least four T gates per successful run, and our protocol attains this bound. The protocol uses 6 to 19 times fewer T gates than gate synthesis at precisions between 10−3 and 10−10. We compare it with a deterministic circuit using an exact repeat-untilsuccess rotation, which achieves a slightly lower expected cost of about 4.67 T gates, and discuss implementation in modular architectures and extensions to N-qubit W states. Note on this version. This version replaces and corrects the first version of this preprint (“A Tcount Zero Protocol for Heralded W-state Preparation via Clifford-only Gates”). The claim there that the W state can be prepared with Clifford gates alone is incorrect: as shown here, some non-Clifford step is necessary, and at least four T gates are needed per successful run.

DOI
10.21203/rs.3.rs-9712730/v1

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Abstract The W state is a key resource for quantum communication and distributed quantum computing. Its entanglement survives the loss of any single qubit. Preparing the W state exactly is nevertheless hard in fault-tolerant architectures. The amplitude 1/√3 lies outside the ring Z[1/√2, i]. So no Clifford+T unitary can prepare the three-qubit W state exactly, and deterministic circuits must approximate the required rotation at a T-count that grows as log(1/ϵ). We present a heralded protocol that prepares the three-qubit W state exactly using Clifford gates and a single non-Clifford operation. A shallow Clifford circuit with two CNOT gates first prepares an equal superposition of |001⟩, |010⟩, |100⟩ and |111⟩. A logical-AND measurement on one ancilla then removes the |111⟩ component without disturbing the remaining coherent superposition. The protocol succeeds with probability 3/4, a failed attempt leaves the known product state |111⟩, and each attempt consumes four T gates or one |CCZ⟩ magic state. This gives an expected cost of about 5.33 T gates independent of target precision. We prove that some non-Clifford step is necessary, and by exhaustively enumerating all 315,057,600 six-qubit stabilizer tensors we show that no post-selected Clifford circuit with three or fewer T gates outputs the W state. Every heralded Clifford+T preparation of the W state therefore uses at least four T gates per successful run, and our protocol attains this bound. The protocol uses 6 to 19 times fewer T gates than gate synthesis at precisions between 10−3 and 10−10. We compare it with a deterministic circuit using an exact repeat-untilsuccess rotation, which achieves a slightly lower expected cost of about 4.67 T gates, and discuss implementation in modular architectures and extensions to N-qubit W states. Note on this version. This version replaces and corrects the first version of this preprint (“A Tcount Zero Protocol for Heralded W-state Preparation via Clifford-only Gates”). The claim there that the W state can be prepared with Clifford gates alone is incorrect: as shown here, some non-Clifford step is necessary, and at least four T gates are needed per successful run.
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Approximate synthesis of general single-qubit unitaries over the Clifford+T\sqrt{T}T​ gate set
For the standard Clifford+TTT gate set, deterministic, ancilla-free synthesis now attains the minimal TTT-count for general single-qubit unitaries (Morisaki et al., arXiv:2510.05816). The T\sqrt{T}T​ gate rotates by half the angle of TTT, generating a finer lattice of implementable operations. Access to this magic state is expected to lower the cost of deterministic and ancilla-free synthesis of general single-qubit unitaries (Kliuchnikov et al., arXiv:2203.10064), but no direct Clifford+T\sqrt{T}T​ algorithm existed for this case. We provide one by extending the integer lattice-point enumeration method of Morisaki et al. We adopt a resource state cost model based on the magic-state catalysis approach of Gidney and Fowler (arXiv:1812.01238). On Haar-random targets synthesized to precisions ranging from ε=10−3\varepsilon=10^{-3}ε=10−3 to 10−810^{-8}10−8, the cost of Clifford+T\sqrt{T}T​ circuits scales as 2.4log⁡2(1/ε)2.4\log_2(1/\varepsilon)2.4log2​(1/ε) compared to 3.0log⁡2(1/ε)3.0\log_2(1/\varepsilon)3.0log2​(1/ε) for TTT-count-optimal Clifford+TTT circuits. Once a one-time catalyst state is amortized, the Clifford+T\sqrt{T}T​ circuits are never costlier than their Clifford+TTT counterparts.
Preparing a quantum circuit that implements a given sparse state is an important building block that is necessary for many different quantum algorithms. In the context of fault-tolerant quantum computing, the so-called non-Clifford gates are much more expensive to perform than the Clifford ones. We hence provide an algorithm for synthesizing sparse quantum states with a special care for quantum resources. The circuit depth, ancilla count, and crucially non-Clifford count of the circuit produced by the algorithm are all linear in the sparsity when access to arbitrary-angled rotations is given. When compiled down to the standard Clifford+T gate set, several constructions can be given for increasingly better T-count and depth at the expense of a larger number of ancillae. The most optimised construction for T-count reaches O(s+log⁡2(1/ε))\mathcal O\left(s+\log_2(1/ε)\right)O(s+log2​(1/ε)) T gates for error εεε. The constructions are broken into two parts, one that synthesises a generalized W-state, well studied in the literature; and the second which is a classical reversible circuit implementing a permutation that maps the basis states of the W-state to those of the target sparse quantum state. We reduce this problem to the diagonalization of a binary matrix, using a specific set of elementary matrix operations corresponding to the classical reversible gates. We then solve this problem using a new version of Gauss-Jordan elimination, that minimizes the circuit complexities including circuit depth using parallel elimination steps. When the circuit is applied in one direction, we notice that all occurrences of (the expensive) Toffoli gates can all be replaced by adaptive Clifford circuits, leading to a better non-Clifford count.
Magic state distillation is one of the leading candidates for implementing universal fault-tolerant logical gates. However, the distillation circuits themselves are not fault-tolerant, so there is additional cost to first implement encoded Clifford gates with negligible error. In this paper we present a scheme to fault-tolerantly and directly prepare magic states using flag qubits. One of these schemes requires only three ancilla qubits, even with noisy Clifford gates. We compare the physical qubit and gate cost of our scheme to the magic state distillation protocol of Meier, Eastin, and Knill (MEK), which is efficient and uses a small stabilizer circuit. For low enough noise rates, we show that in some regimes the overhead can be improved by several orders of magnitude compared to the MEK scheme which uses Clifford operations encoded in the codes considered in this work.
Efficient synthesis of arbitrary quantum states and unitaries from a universal fault-tolerant gate-set e.g. Clifford+T is a key subroutine in quantum computation. As large quantum algorithms feature many qubits that encode coherent quantum information but remain idle for parts of the computation, these should be used if it minimizes overall gate counts, especially that of the expensive T-gates. We present a quantum algorithm for preparing any dimension- N pure quantum state specified by a list of N classical numbers, that realizes a trade-off between space and T-gates. Our scheme uses O ( log ⁡ ( N / ϵ ) ) clean qubits and a tunable number of ∼ ( λ log ⁡ ( log ⁡ N ϵ ) ) dirty qubits, to reduce the T-gate cost to O ( N λ + λ log ⁡ N ϵ log ⁡ log ⁡ N ϵ ) . This trade-off is optimal up to logarithmic factors, proven through an unconditional gate counting lower bound, and is, in the best case, a quadratic improvement in T-count over prior ancillary-free approaches. We prove similar statements for unitary synthesis by reduction to state preparation. Underlying our constructions is a T-efficient circuit implementation of a quantum oracle for arbitrary classical data.
Clifford+T synthesis of rotation gates is an important routine in fault-tolerant quantum compilation. While Clifford+T synthesis is scalable, it has a high overhead of tens of T gates per rotation in practice, translating to high resource estimates for many fault-tolerant algorithms. However, these well-known results, including those using probabilistic mixtures [Quantum 7, 1208 (2023)], are independent of the rotation angle θθθ, requiring O(log⁡1/δ)O(\log 1/δ)O(log1/δ) T gates. We show that it is possible to do much better for small angles, reducing the T cost to O~(θ2/δ)\tilde{O}(θ^2/δ)O~(θ2/δ), and returning to existing O(log⁡1/δ)O(\log1/δ)O(log1/δ) results in the worst case. This is particularly important since many algorithms, such as Trotterization, are dominated by small-angle rotations. Further, we perform a detailed theoretical and numerical study of quasi-probabilities, which can further reduce the total T cost of large circuits by orders of magnitude with only a small overhead in sample complexity. We also develop a scheme based on quasi-probability mixtures of Clifford+T fallback channels. We derive new θθθ-dependent formulas that can be used for resource estimation of fault-tolerant quantum algorithms. As an application of our results, we show that the gate cost of Trotterization circuits compiled to a Clifford+T gate set is constant in the small Trotter step size limit, and can be reduced by orders of magnitude even for large step sizes. The cost of fault-tolerant Trotterization for a variety of applications should be re-examined in light of these results. Our work dispels the widely-stated claim that Clifford+T rotation synthesis has a high cost independent of θθθ, and further develops a scalable quasi-probability method for rotation synthesis. We also expect our results to bring forward useful early fault-tolerant quantum computing by reducing required magic state resources.